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concepts at the same level in the hierarchy) are bigger. The second difference lies in
the opacity of presented nodes. Due to the reduction process presented below, we
know that some concepts are not fully reliable (concepts containing objects with
attributes modified during the reduction).
3.2.2 Nonnegative Matrix Factorization
The matrix factorization methods decompose one - usually a huge - matrix into
several smaller ones. Nonnegative matrix factorization differs in that the use of
constraints produces nonnegative basis vectors, which make possible the concept of
a parts-based representation. Common approaches to NMF obtain an approxima-
tion of V by computing a ( W , H ) pair to minimize the Frobenius norm of the
difference V
R mn be a nonnegative matrix and W
R mk and
WH . Let V
2
2
R kn
H
2
for 0
<
k
min
ð
m
;
n
Þ
. Then the objective function or minimization
problem can be stated as
2
min V
k
WH
k
;
with W ij >
0 and H ij >
0
;
for each i and j
:
There are several methods for computing NMF. We have used the multiplicative
method algorithm proposed by Lee and Seung [ 26 , 27 ].
3.2.3 Lorenz Curves
To evaluate similarity we can use the Lorenz curves, an approach well-known in
the economy sector, in the way proposed in [ 14 ]. Let us suppose we have two
presence-absence (binary) arrays r
( y i ) 1, ... , N of dimen-
sion N . In the same manner as we normalize vectors, we can create arrays a i and b i
by dividing each element of the array by their total sum. Formally a i ¼
¼
( x i ) 1, ... , N and s
¼
x i
T r ;
, where T r ¼ P 1 x j and T s ¼ P 1 y j . Next, we can
compute difference array d
y i
b i ¼
T s ; 8
i
¼
1
; ...;
N
;
¼
( d i ) i ¼
1,
...
, N as d i ¼
a i
b i , ordered from the
largest value to the smallest one. By putting c i ¼ P 1 d j , we obtain the coordinates
of the Lorenz similarity curve by joining the origin (0, 0) with the points of
coordinates
c i 1 ;...; N .
N ;
i
3.3 Experiments
3.3.1 Real-World Experiment
In our first example, we will use the well-known dataset from [ 11 ]. We have been
dealt with this experiment previously in [ 35 ] and basically this dataset has the same
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